Fatou's theorem

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In mathematics, specifically in complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk.

Motivation and statement of theorem

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If we have a holomorphic function   defined on the open unit disk  , it is reasonable to ask under what conditions we can extend this function to the boundary of the unit disk. To do this, we can look at what the function looks like on each circle inside the disk centered at 0, each with some radius  . This defines a new function:

 

where

 

is the unit circle. Then it would be expected that the values of the extension of   onto the circle should be the limit of these functions, and so the question reduces to determining when   converges, and in what sense, as  , and how well defined is this limit. In particular, if the   norms of these   are well behaved, we have an answer:

Theorem. Let   be a holomorphic function such that
 
where   are defined as above. Then   converges to some function   pointwise almost everywhere and in   norm. That is,
 

Now, notice that this pointwise limit is a radial limit. That is, the limit being taken is along a straight line from the center of the disk to the boundary of the circle, and the statement above hence says that

 

The natural question is, with this boundary function defined, will we converge pointwise to this function by taking a limit in any other way? That is, suppose instead of following a straight line to the boundary, we follow an arbitrary curve   converging to some point   on the boundary. Will   converge to  ? (Note that the above theorem is just the special case of  ). It turns out that the curve   needs to be non-tangential, meaning that the curve does not approach its target on the boundary in a way that makes it tangent to the boundary of the circle. In other words, the range of   must be contained in a wedge emanating from the limit point. We summarize as follows:

Definition. Let   be a continuous path such that  . Define

 

That is,   is the wedge inside the disk with angle   whose axis passes between   and zero. We say that   converges non-tangentially to  , or that it is a non-tangential limit, if there exists   such that   is contained in   and  .

Fatou's Theorem. Let   Then for almost all  
 
for every non-tangential limit   converging to   where   is defined as above.

Discussion

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See also

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References

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  • John B. Garnett, Bounded Analytic Functions, (2006) Springer-Verlag, New York
  • Krantz, Steven G. (2007). "The Boundary Behavior of Holomorphic Functions: Global and Local Results". Asian Journal of Mathematics. 11 (2): 179–200. arXiv:math/0608650. doi:10.4310/AJM.2007.v11.n2.a2. S2CID 56367819.
  • Walter Rudin. Real and Complex Analysis (1987), 3rd Ed., McGraw Hill, New York.
  • Elias Stein, Singular integrals and differentiability properties of functions (1970), Princeton University Press, Princeton.